Terrain is a multi-scale (or even fractal1) phenomenon. Thus, with Wood (1996)2 we can state:
It would seem ludicrous to only consider surface variation at a fixed scale when an assessment of an entire landscape is desired. Our own judgements both scientifically and “intuitively” rely on an appreciation of landscape at a variety of scales simultaneously.
Thus, it is interesting that ArcGIS Pro 3.4 and 3.5 add functions for computing the multi-scale surface percentile, the multi-scale surface difference, and the multi-scale surface deviation, respectively.
More advanced geomorphometric parameters such as multi-scale surface shape classification, various kinds of multi-scale curvatures, and more can be computed using SAGA GIS (e.g. through QGIS with the appropriate plug-in) or using LandSerf by the afore-mentioned Jo Wood.
Footnotes
The fractal dimension of a terrain surface can be estimated (similarly to the more famous example regarding the fractal dimension of coastlines). In fact, I have done so for Switzerland using Swisstopo DEMs years ago. But that nugget of information is lost to the implosion of Twitter.↩︎
Transparency note: Link points to an article by myself on the blog of my employer.↩︎